pha aah Lae | Coast Eng. Res. Ctr. CETA 81-2
Seaward Limit of Significant Sand Transport by Waves: An Annual Zonation for Seasonal Profiles
by
Robert J. Hallermeier
COASTAL ENGINEERING TECHNICAL AID NO. 81-2
JANUARY 1981
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REPORT DOCUMENTATION PAGE BERGHE ICONEDE aAGInOEe
1. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER CETA 81-2
4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED
SEAWARD LIMIT OF SIGNIFICANT SAND TRANSPORT BY Cogisitcall Bmginecrang
WAVES: AN ANNUAL ZONATION FOR SEASONAL PROFILES teelnaenty aug 6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(a)
Robert J. Hallermeier
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK Department of the Army AREA & WORK UNIT NUMBERS
Coastal Engineering Research Center (CERRE-CS) D31193 Kingman Building, Fort Belvoir, Virginia 22060
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army January 1981 Coastal Engineering Research Center aeINOMBERIORIDAGES Kingman Building, Fort Belvoir, Virginia 22060 23 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED
15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE
DISTRIBUTION STATEMENT (of this Report)
Approved for public release, distribution unlimited.
DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES
- KEY WORDS (Continue on reverse side if necessary and identify by block number)
Beach profiles Coastal engineering Sand transport Wave climate application Beach zones Coastal processes Seaward limit Wave-sand interactions
20. ABSTRACT (Continue on reverse side if necesaary and identify by block number)
Sand characteristics and annual wave statistics at a site are used to deter- mine two water depths bounding a shoal zone on the beach profile. This zonation is based on two thresholds of wave-induced sand agitation, so that expected waves during a year have neither strong nor negligible effects on the sand bottom with- in the shoal zone. The calculation procedure and representative results for the shoal zone bounds are presented to supplement techniques for estimating a seaward limit of significant sand transport given in the Shore Protection Manual (SPM).
A calculator program is provided.
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PREFACE
This report provides a new technique for seaward-limit estimation, supple- menting those given in Section 4.523 of the Shore Protection Manual (SPM). Two calculated water depths locate the edges of a buffer region called the shoal zone, which is defined so that significant alongshore transport and intense on- shore-offshore transport by waves occur landward of this zone, and only insig-— nificant onshore-offshore transport by waves occurs seaward of this zone. This profile zonation and guidance on its applications are interim results from an ongoing study conducted within the beach behavior and restoration research pro- gram of the U.S. Army Coastal Engineering Research Center (CERC).
The report was written by Dr. Robert J. Hallermeier, Oceanographer, under the general supervision of C. Mason, Chief, Coastal Processes Branch, and then Dr. R.M. Sorensen, Chief, Coastal Processes and Structures Branch.
Comments on this publication are invited.
Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963.
D E. BISHOP
Colonel, Corps of Engineers Commander and Director
APPENDIX
CONTENTS
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) .
SYAUISOILS, AND) IDIETONIOIIIONISS 6 6 a 06 oo Oo OOO Oo JONURODIUGIILONN6 6 6 66 5 06 Oo Oooo DO OO BASIC EQUATIONS AND SOLUTION TECHNIQUES ...... NASI MUONS Oi cig AND G4o 6 6 6 6 6 6 4 66 oo 6
SHOAL ZONE EXTENT ALONG U.S. COASTS ....... =.
SUMMARY] ep aot Ratner coy (oho oti ar tant ote tan RMN URIS Tuan strc Ls =
IIIT, (IIS 6 5 6 6 6 6 6 65 6 6 6 oe 6 OO
CALCULATOR PROGRAM FOR THE BOUNDS OF THE SHOAL ZONE
TABLES
1 Calculation procedure for locating bounds of the shoal zone.
2 Calculated shoal zone bounds for 10 U.S. sites using annual
statistics of nearshore wave gage measurements. ....
3. Calculated shoal zone bounds for 20 U.S. sites using annual
SALEM SECS Oe IO) lpeeeleeie OIMGGAVEIETIONS 4 5 6 6 5b 0 0 oO Oo
1 Definition sketch of annual sand beach zonation and terminology. .
2 Graphical definition of statistics of exponential cumulative
FIGURES
AWE InGieine chigtieiiMmirte@ne 6 6 6 6 6 o 3. Graphical solution of equation (5) for y' =1.6...... 4 Graph of hyperbolic sine and tangent functions ......
e
o ile
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows:
Multiply by To obtain inches 2504 millimeters 2.54 centimeters Square inches 62452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 Square meters cubic feet 0.0283 cubic meters yards 0.9144 meters Square yards 0.836 Square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259-0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars NOI? x LOT” kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angel) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins
To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32).
To obtain Kelvin (K) readings, use formula:
K = (5/9) CF =32) a 2730150
SYMBOLS AND DEFINITIONS median sand grain diameter water depth water depth at seaward bound of shoal zone water depth at landward bound of shoal zone acceleration of gravity wave height annual mean significant wave height extreme significant wave height exceeded 12 hours per year annual median significant wave height linear wavelength linear wavelength in deep water wave period annual mean significant wave period maximum linear wave-induced horizontal velocity near the bed argument of hyperbolic functions in Figure 4 ratio of density difference between sand and fluid to fluid density
annual standard deviation of significant wave height
SEAWARD LIMIT OF SIGNIFICANT SAND TRANSPORT BY WAVES: AN ANNUAL ZONATION FOR SEASONAL PROFILES
by Robert J. Hallermeter
I. INTRODUCTION
Certain coastal activities require consideration of the seaward limit to significant wave-induced bed activity along a sand beach profile. Examples in- clude design of nearshore structures, subaqueous beach fill, and borrow or disposal of material. The seaward limit to significant sand movement by waves clearly depends on both wave and sand characteristics through the mechanisms of wave agitation of sand. However, available guidelines on the seaward limit (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977, pp. 4-63 to 4-70) proceed from indirect or qualitative consideration of wave-sand interactions. This report provides a new quantitative procedure for seaward- limit estimation based on definite wave-sand interactions. A comprehensive discussion of the present procedure and of other seaward-limit treatments is provided in Haliermeier (1981).
The present procedure defines an annual beach profile zonation related to the seaward limit. Figure 1 is a schematic illustration of this zonation and identifies the terminology used here. The water depths dg and dy, respec- tively the landward and seaward bounds of the shoal zone, are calculated in Section II, using sand characteristics and statistics of annual wave climate for a given locality. This shoal zone is a buffer area where expected waves have neither strong nor negiigible effects on the sand bed during a typical annual cycle of wave action, according to availabie knowledge of sand movement by waves.
The water depth, d,, gives a seaward limit to extreme surf-related effects, so that significant alongshore transport and intense onshore-offshore transport are restricted to-water depths less than d,. The water depth, Gla 5 gives a seaward limit to sand motion by usual waves, so that significant onshore-offshore transport is restricted to water depths less than d,;. The seaward limit in sandy regions reasonably falls within this calculated shoal zone, although place- ment of the limit depends on the particular engineering application, as pointed out in Section III.
Seasonal Range of Sand Level
Tidal Range
mE
— Mean Seo Level
—Meon Sand Level 1
OFFSHORE Di SHOAL >| LITTORAL ZONE ZONE ZONE
Figure 1. Definition sketch of annual sand beach zonation and terminology.
Since the shoal zone has a definite physical basis, the present calculation procedure may reduce the need for detailed site-specific investigations of phys- ical processes, and certainly aids in designing such investigations at localities where wave climate is known. Nearshore measurements of wave climate along U.S. coasts provide a basis for the example calculations given in Section IV. The calculated shoal zone extent is to be considered in conjunction with other in- formation pertinent to the seaward limit of the wave-dominated sand profile.
II. BASIC EQUATIONS AND SOLUTION TECHNIQUES
The water depths dg and dj are calculated from relationships giving the threshold flow energy for two distinct wave interactions with a sand bed. The major assumptions involved are the accuracy of linear wave theory for calcula- tions of near-bottom maximum flow velocity, and a modified exponential distri- bution of nearshore wave heights.
The landward bound to the shoal is determined from the relationship
2 “Bex Ca) = 0.03 (1) where Unax(-d) = the maximum wave-induced horizontal fluid velocity near the bed Vv = the ratio of the density difference between sand-and fluid to the fluid density
g = the acceleration due to gravity d = the mean water depth.
Equation (1) describes a definite threshold of erosive sand-bed agitation by steady wave action, and upax(-q) will be taken as that due to an extreme wave height exceeded 12 hours per year in determining dg. The water depth then de- fined by equation (1) agrees with field data on the maximum water depth for significant profile changes (elevation excursions greater than +0.5 foot or +0.15 meter) throughout a yearly cycle of wave climate.
Linear wave theory gives
TH
Umax(-d) = T {sinh (2md/L)}
(2)
where H is the wave height, T the wave period, L the local wavelength at d; and
(can Cna7iy | 5 oes (=) (3)
where L, is wavelength in deep water.
The modified exponential distribution approximating typical measured near— shore wave heights is shown in Figure 2. In terms of standard annual wave
x:Measured Cumulative Wave Height Data, 22 Apr. 1948 To 30 Apr. 1949, Long Branch, N.J. (Hall and Herron, 1950)
—
Exponential Regression Line
i
Significant Wave Height (ft)
Cycle Length In 10
i
0.1 Os 5 0 50 100 Pct of Wave Heights Greater Than Ordinate
Figure 2. Graphical definition of statistics of exponential cumulative wave height distribution.
statistics, the extreme significant wave height exceeded 12 hours per year (0.137 percent) is
Heg.ig7 > Hig 7 So @ (4)
where H, is annual average significant wave height, and o is the annual standard deviation of significant wave height. For simplicity, the period as-— sociated with this extreme wave condition is takem to be the average significant wave period Te with fair justification. The wave statistics Hag o, and
T, should be defined by one or more full years of daily or more frequent meas-
urements.
Using equations (2), (3), and (4), equation (1) can be rewritten as
Biss 2nd) \ sinh? J 274, | tann{2™4y J dh (Hy + 5.6 0)? G) L L L 0.03y"(gT2/2m)*
and the values of y', Hg, 6, and T, determine a single value of (2nmdg/L). The
value of dy can be accurately determined using the iterative root-finding cal- culator program presented in the Appendix. Alternatively, for y' = 1.6 (quartz
sand in saltwater), Figure 3 provides a graphical solution of equation (5). The dimensionless value of (2md)/L) is converted to dy, by multiplying by {(Lo/27) tanh (2mdp/L)} (see eq. 3); Figure 4 provides the needed graph of (tanh x) ver- sus x. Another alternative for y' = 1.6 is the first-order analytical approxi-
mation,
dy = 2H. ar il G, (6) which is convenient and usually within +5 percent of an exact calculation.
The seaward bound to the shoal zone is determined from the relationship
72 Unax (-d) ? = 8, y gD W)
where D is the median sand diameter characteristic of the shoal zone bed, e.g., at mean water depth of (1.5 dy). Equation (7) gives the threshold of sand motion by wave action in field conditions, and Umax(-d) will be taken as that due to
the median (50 percent) wave condition. For this wave condition, equation (7) along with the depth dependence in equation (2) defines a maximum water depth
for sand motion, d;, which corresponds to the seaward limit of the usual wave- constructed profile.
The modified exponential distribution gives the median significant wave height as
Hssq = Hs - 0.3 o (8)
The period associated with this median wave height is also taken to be the average wave period T,. With these relationships, it follows that
2nd; o-(@. = O23 @)>|| V2 oe) sinh = ear L 8y'gDTs
The value of dy can be accurately determined using the calculator program pre- sented in the Appendix. Alternatively, after calculating the value of the right side of equation (9), the value of (21d;/L) can be determined from the graph of x versus (sinh x) in Figure 4, and converted to dj by multiplying by {(Lo/2m) tanh (2nd;/L)}, using the Figure 4 graph of (tanh x) versus x. Another alter-
native, for y' = 1.6, is to use the empirical approximation
~
da = (ey > 0.3 G) Be G/5000 w)%>° (10) which is convenient and usually within +5 percent of an exact calculation.
Table 1 summarizes the basic procedure for calculating the water depths at the two bounds of the shoal zone, for specified values of g, y', D, Hee G5, anal T,. The procedure can be executed using either the calculator program provided in the Appendix or the graphs provided in Figures 3 and 4. A convenient alter- native is to usé the simple approximations in equations (6) and (10), which
10
no
1.0
5 © D> FD
o a
°S rh
Dimensionless Depth, 2™dg/L (oe) BS
0.1 100 5.410° 10° 5.107 10" Extreme Wave Steepness, {Hs orsr/t¢ = {iis #5601/1075%2 nh} Pines J, Crejiwleal solpicion ot equation (5) fore y! = 1.6,
Cam ifieek. . n
Hyperbolic Function of X os p
—
°0 0.5 1.0 1.5 2.0 2.9
Argument, X Figure 4. Graph of hyperbolic sine and tangent functions.
11
Table 1. Calculation procedure for locating bounds of the shoal zone.
Required site parameters: y' and Hg, 6, Ts, and g in consistent units.
Find dimensionless root (21dg/L) of equation (5), either by iteration using the calculator program in the Appendix or from Figure 3 after cal- culating the value of the abscissa {27 (Hy + 5.6 o)/gTS}.
Calculate a 2nd 2nd gT
aa = (“*)feam (I). L L 4n2
Additional required site parameter: D in consistent units at water depth of (1.5 dg).
Seaward
Find the dimensionless value of (2ndy/L) specified by equation (9), using either the calculator program in the Appendix or Figure 4 after calcuiating the value of the ordinate
n2 (Hg - 0.3 oa 8y'gDTZ
Calculate
provide results usually consistent in accuracy with the critical values given in equations (1) and (7) to one significant digit. To allow for possible effects of tides, it is recommended that the calculated dg and dj be taken as being with respect to mean lower low water (MLLW) at a locality, giving conservative (farther seaward) locations for the shoal zone boundaries. Also, it seems ap- propriate to use calculated results in U.S. customary units (feet), rounded to the nearest integer unit.
III. APPLICATIONS OF dy AND dj
The shoal zone is an objectively defined buffer area of moderate wave effects on the submerged beach profile throughout the year. Where wave and sand charac- teristics are known, the two calculated water depths are useful in various coastal engineering activities requiring a seaward-limit estimate. The possible applications in the following list are discussed in detail in Hallermeier (1981), and further evaluations of shoal zone uses are being conducted.
(a) Fathometer surveys of customary vertical resolution (+0.5 foot or +0.15 meter) may not be expected to resolve usually small bottom changes during the year seaward of dg. Nearshore surveys extending to dy are needed to define seasonal conditions at a site.
(b) Gages to record waves incident on the littoral zone may be placed just seaward of the anticipated value of dy, so that the meas— ured waves will not be modified differently through the year due to significant changes in the bottom seaward of the gage site.
12
Wasi
(c) Sand for beach fill must be placed landward of dg to in- crease the sand supply in the intensely active littoral zone. The necessary quantity of nourishment material is at least that required to advance the average profile landward of dp the desired distance.
(d) In calculations of shoreline erosion due to sea level rise over long timespans, the ocean boundary to the equilibrium nearshore
profile may be set at dj.
(e) In calculations of sediment budgets pertaining to timespans less than a few decades, the ocean boundary to the control volume may be set at dg.
(f£) At sites with shore-parallel contours, subaqueous borrow or disposal of material should be conducted well seaward of dy, so that destructive effects on nearshore processes are minimized. Sea- ward of dg, borrow or disposal might be conducted in a region with irregular contours if bottom elevations and thus shore exposure are not changed significantly.
(g) A rubble-mound breakwater should be sited in water deeper than dg for the proximate region, if its primary purpose is to provide wave shelter with minimum effect as a littoral barrier.
(h) To balance economy and function, it may be advisable to build a shore-normal structure to the length corresponding to the deg con- tour for the nearby region if its primary purpose is to control littoral drift.
IV. SHOAL ZONE EXTENT ALONG U.S. COASTS
Tables 2 and 3 present bounding water depths for the shoal zone at selected coastal localities. Using the calculator program provided in the Appendix,
calculations for the 10 sites in Table 2 use annual summary statistics from
the wave climate data reported by Thompson (1977). For these sites, at least
1 full year of nearshore surface-piercing gage data is available, with summary
statistics provided by objective digital record analysis.
Table 2. Calculated shoal zone bounds for 10 U.S. sites using annual statistics of nearshore wave gage measurements (Thompson, 1977); y' = 1.6, g = 32.2 feet per second squared (1 foot = 0.305 meter, 1 millimeter = 0.0033 foot).
Location
Atlantic coast
Atlantic City, N.J. Virginia Beach, Va.
Nags Head, N.C.
Atlantic Beach, N.C. Wrightsville Beach, N.C. Hoiden Beach, N.C.
Lake Worth, Fla.
Gulf of Mexico coast
26°08' N., 81°49' w. 30°23' N., 86°25" w.
Naples, Fla.
Destin, Fla.
Pacific coast
Huntington Beach, Calif.
33°39" 1.,118°00'
1
Table 3. Calculated skoal zone bounds for 20 U.S. sites using annual statistics of LEO breaker observations; y' = 1.6, g = 32.2 feet per second squared, D = 0.00033 foot (1 foot = 0.305 meter).
Atlantic coast Assateague, Md. 38°11' N., 75°09" w.| 1978 ; ; 5 73.3 Bull Island, S.C. 32°55' N., 79°35' w.| 1977-78] 2. i ; Z 52.7
N., 80°50’ W.| 1976-77] 2. 5 : h 68.7
N., 80°04" w.| 1971 ; . : .7] 31.2
Tybee Lighthouse, Ga. Boca Raton, Fla.
Gulf of Mexico coast
St. Andrews Park, Fla. N., 85°40' w.| 1969-70] 1. 3 : .7| 23.6 Crystal Beach, Fla. N., 86°27" w.] 1969-70] 1. : off 81) Aoi Gilchrist, Tex. N., 94°29" w.| 1975 ; ; ay 5B) Blo Galveston, Tex. N., 94°58" w.} 1975 0 6 9 5 39.6 Corpus Christi, Tex. N., 97°10" w.| 1974 : u : 4 6C.9
Pacific coast 5 MPH ., 118°02' ., 119°09' ., 120°39' ., 121°07' 121°56' ., 122°28" , 123°06' , 124°04" , 124704! , 124°13'
San Clemente, Calif. Bolsa Chica, Calif. Pe. Mugu, Calif.
Pismo Beach, Calif. San Simeon, Calif. Capitola Beach, Calif. Stinson Beach, Calif. Wright's Beach, Calif. Shelter Cove, Calif. Prairie Creek, Calif.
ES agh OES SEH ead Eat ES SE Fal)
EH 4h ee Ses oS GR ES ESA ES OS Dn ON OD DON DDO FF Ww oO Cc
wBwBwn Ff Ww KF WwW WwW DY DY DH FPF Ry FP OF RP Be ee
Umpqua, Oreg.
Calculations for the 20 sites in Table 3 use annual summary statistics of breaker observations collected under the LEO program conducted by the U.S. Army Coastal Engineering Research Center (CERC). Such wave data are liable to criti- cism regarding precision and subjectivity, and seem to give larger annual mean wave heights than gages located seaward of the littoral zone. Also, for the Table 3 calculations, D has been set equal to 0.00033 foot (0.1 millimeter),
a typical value. However, the Table 3 results show consistency with Table 2 results for common regions.
kA RK KK kK kk & OK O&K O&K K K EXAMPLE PROBLEM * * * *& & kK KKK KKK KK KK
GIVEN: At Galveston, Texas (29°17" No 5 94°47" W.), annual wave statistics for 1966 (Thompson, 1977) are:
Mean significant height, He = 1.35 feet Standard deviation of height, o = 0.9 foot Mean wave period, TZ = 5.7 seconds.
Median sediment diameter at the 18-foot water depth off Bolivar Peninsula just north of Galveston (U.S. Army Engineer District, Galveston, 1958) is:
D = 0.00033 foot (0.1 millimeter).
FIND: The water depths d, and d;, giving the extent of the shoal zone at
this site. SOLUTION: From equation (4),
Hen ay = lose =P Goce) OsO)} = G8 ask,
and from equation (3),
325 21Gabs
= 166.5 feet 20
me
so that
and Figure 3 gives
Figure 4 gives tanh (0.74) = 0.63,
so that
21d IE 21d ave ( q £)(F2), conn ( q a = (0.74) (26.5) (0.63) = 12.4 feet = 12 feet. L T L
Note that equation (6) gives
dy = {2.35 + 11(0.9) |} = 12.6 feet | 13 fest.
From equation (8) ,
Higea = les = On3@.9)) = 1003 secs,
Ss
so equation (9) becomes
‘
sani ( wt). i = 1.61 iL 3.6) G22) O.0U033)(Ga2)-
Figure 4 gives
21d sinh7!(1.61) = 1.26 -( = 3
and
so that
2nd; 1b; 21d. dj = ( L )(oa) tanh [4 = (GL 526) (20,45) (585) = 28.33 teee © 28 tear.
Note that equation (10) gives
d; = {(1.08) (5.7) [32.25/5,000(0.00033) ]9-5} = 27.2 feet = 27 feet.
With the same input parameters, the calculator program in the Appendix gives the more precise results:
Gly SUAS) Hest S 2 eee
d; = 28.32 feet = 28 feet.
Hey Wie He OR URRY eae. utes be Re ieee: aes deluded cae: cde Fea de) ide! ode adel eee oe eC a
These calculated results are to be considered in conjunction with other possible indicators of the seaward limit to the wave-dominated profile. Everts (1978) reported two geometric limit depths for Palm Beach near the center of Galveston Island (29912' N., 94°57' W.). The limit depth to shore-parallel bathymetry is 47 feet (14.3 meters), and the depth at the boundary between the shoreface and ramp profile sectors is 59 feet (18.0 meters); both depths are with respect to mean low water (MLW).
Sand-size variation along the profile can indicate a seaward limit to wave effects (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977). For the Galveston area, Fisher, et al. (1972) charted the boundary be- tween "shoreface sand and muddy sand" and "shelf mud and sand with shell" at about 28 to 29 feet below mean sea level (MSL). (MLW is about 1 foot below MSL at this site.)
Watson and Behrens (1976) presented six profiles surveyed to about 14 feet below MSL in June and December 1974 and March 1976 near New Corpus Christi Pass, Texas (28°41' N., 97°1l' W.). Five of these profiles superpose to within +0.5 foot beyond a water depth of about 10 to 12 feet below MLW. Charted profile geometry at this site is similar to that near Galveston (Everts, 1978), and wave climate may be expected to be similar.
The preceding information supports the calculated values of d,; and dy for this site. In this example, the shoal zone seems well-founded, while the two geometric indicators derived by Everts (1978) are not supported by other evi- dence on the limits to wave effects on the submarine profile.
V. SUMMARY
This report is a self-contained presentation of the calculation procedure (Table 1) for locating a shoal zone along a sand beach profile. The water depths bounding this shoal zone depend on sand characteristics (y", D) and sum- Mary statistics of annual wave climate (a, o, and ‘s)) c The shoal zone is de- fined so that expected surface waves have only moderate effects on the sand bottom in this zone throughout a typical year. The calculated shoal zone bounds have applications in coastal engineering activities requiring an estimate of the seaward limit to the wave-dominated nearshore region (see Sec. III).
16
The assumptions involved in the shoal zone model are limiting, to some ex— tent, and confirmatory field evidence is definitely meager, but the present calculation procedure is objective, and seems to be a useful new tool for engi- neering applications. The calculated shoal zone supplements other estimation techniques for the seaward limit (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977); of these other techniques, analysis of sediment character variation along the profile seems the most defendable.
LITERATURE CITED
EVERTS, C.H., "Geometry of Profiles Across Inner-Continental Shelves of the Atlantic and Gulf Coasts of the United States," TP 78-4, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Apr. 1978.
FISHER, W.L., et al., "Environmental Geologic Atlas of the Texas Coastal Zone, Galveston-Houston Area," Bureau of Economic Geology, University of Texas, Mugen, WE 4 IDA.
HALL, J.V., Jr., and HERRON, W.J., "Test of Nourishment of the Shore by Offshore Deposition of Sand," TM-17, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., June 1950.
HALLERMEIER, R.J., "A Profile Zonation for Seasonal Sand Beaches from Wave Climate," to be published in Coastal Engineering, 1981.
KOMAR, P.D., "Boundary Layer Flow Under Steady Unidirectional Currents," Marine Sediment Transport and Environmental Management, John Wiley & Sons, Inc., New York, 1976.
THOMPSON, E.F., “Wave Climate at Selected Locations Along U.S. Coasts," TR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Jan 1977.
U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Proteetton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.
U.S. ARMY ENGINEER DISTRICT, GALVESTON, ''Report on Beach Erosion Control Cooperative Study of the Gulf Shore of Bolivar Peninsula, Texas," Galveston, Tex., May 1958.
WATSON, R.L., and BEHRENS, E.W., "Hydraulics and Dynamics of New Corpus Christi Pass, Texas: A Case History, 1973-75," GITI Report 9, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., and U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., Sept. 1976.
APPENDIX CALCULATOR PROGRAM FOR THE BOUNDS OF THE SHOAL ZONE
A calculator program executing the Table 1 procedure is documented on the following four pages. Written for the Hewlett-Packard HP-6/ Programmable Pocket Calculator, this program uses metric or U.S. customary units, 224 program steps, 25 storage registers, and RPN logic. The program could be translated for use with calculators having other logic systems and similar features and capacities.
The program incorporates the basic relationships in equations (1) and (7), the needed identities for hyperbolic functions and for linear waves, and an effective interative root-finding routine. The parameters to be specified for the calculations are Bla O, Ps D, y', and g. For quartz sand, y' = 1.6 in seawater and y' = 1.65 in freshwater, with slight variations with water tem- perature and, of course, salinity (Komar, 1976). For most sites, especially at sea level in temperate latitudes, the value of g may be taken to be 32.2 feet (981 centimeters) per second squared.
Program Description
Program Title Bounding Water DEPTHS of Wave-Basep SHoar towe
Name Rowpeet J. HarlrermeieR Date Jan. 1981 Address Researcn Division, U.S. Army Coastar Enaineéeewe (esearcu Center City Kingman Buiping, Foer BervoiR — state VinGinia Zip Code 22066
Program Description, Equations, Variables, etc. [HE MAIN PROGRAM PERFORMS AN ITERATIVE SOLUTION OF TWO FUNCTIONS DEFINING DIMENSIONLESS WATER DEPTH AT EACH END 0F A SHOAL’ toNE ALONG THE SUBMERGED BEACH PRoFire. AFTER WAVE AND SAND CHARACTER AT A SITE ARE INPUT, THE SECANT METHOD FINDS A RooT ACCURATE TO THREE SIGNIFICANT FIGURES, OTHER PROGRAMS ; CONVERT THE DIMENSIONLESS RocTS To dq And d; , WATER DEPTHS IN METERS AT THE LANDWARD AND SEAWARD BOUNDARIES , RESPECTIVELY 5 PER ETORM UMTS CONVERSIONS ; * AND CALCULATE CRITICAL VELOCITY , Ney FoR SAND MOTION ENUM; OTHER SY¥MBoLS ARE:
B= RATIO OF DENSITY DIFFERENCE BETWEEN SAND AND FLUID Td FLUID DENSITY: Ts> ANNUAL AVERAGE SIGNIFICANT WAVE PERIOD 5 9 > ACCELERATION OF GRAVITY, = ANNVAL AVERAGE SIGMEICANT WAVE HEIGHT ) = CHARACTERISNC SAND SItEy Oo = ANNVAL STANDARD DEVIATION OF SIGNIFICANT WAVE HEIGHT } L= LOCAL WAVELENGTH; A= NEAR-BoTTOM HORIZONTAL FLOW AMPLITUDE 4 L,= = T*/20)> DEER- WATER WAVE LENGTC 5x = (21d, /L)= DimensionLESs WATER DEPTH, X= DUMMY INDEX STANDING FoR £08 i, Tre Two BASIC FUNCTIONS ARE:
a oe en 3 (Ho S.60)° Sey en oe fecrz] o= %(e ) ‘— So ke ) AND 02 14+f{al(e# -e%)/(H-030)} , Ger 3]
THE SECANT HETROD USES THE ITERATION EQUATION: Se 2 oO) = GG ee) Ure = Ged). Ge 2] OTHER Equations AND DEW mee Scere ee Gn B Hs /2 Shs Gx 8 ne eS aura S(Oe a. 0)” (ior c J en Gaee mes Lo(tanh &,)= L 5 dh 2(E, tanh &) (Lo ee ;
Operating Limits and Warnings ItE INCORPORATED BASIC RESULTS ARE VALID ONLY FoR O?EN- COAST CONDITIONS AND SAND-SIZE SEDIMENTS (D BETWEEN 0.006 AND 0,2 cenTIMETERS), LiveAR WAVE THEORX AND A MODIFIED EXPONENTIAL DISTRIBUTION FOR CUMULATIVE WAVE WEIGHT ARE ASSYMED. THE Roots Sp) AND § CAN BE PHYSICALLY MEANINGFUL ONLY IF THEY ARE BETWEEN 0.1 AND 3.0 , APPROXIMATELY,
20
User Instructions
ROUNDING WATER DEPTHS OF WAVE-BASED, SIAL 2ONE « 2A:landward bund eg CC reritveal 3A\ seaward bond, Gi veloc. fl fa: iF etre Fb: €: 2rmcters calevilation
LoAD PROGRAM INTO CALCULATOR MEMORY
GE: stact
root Finder
SToRE s\TE PARAMETERS :
__A£, NoRMAUTED IMMERSED SEDIMENT DENSITY
+L, ANNUAL AVERAGE SIGNIFICANT WAVE TERUOD
| STO [STO
Aik» ANNUAL AVERAGE SIGNIFICANT WAVE HEIGHT
STO STo
inv. ANNUAL STANDARD DEVIATION OF HEIGHT
STO
STO
_w. AcceLve Raton OF GRAVITY.
STO
Tr ANY INPUT PACAMETER 1S IN NON-METRIC
UNITS AND Jor. ouTPUTS ARE WANTED IN No
Sere cae CONVERSION PROGRAM .
ne 0. 6 AND BEGIN ITERATION,
WAIT FOR STATIONARY DISPLAY OF §y RooT. ConvERT §, To dy IN METERS AND STORE.
Te vested, convert dy to FEET.
EXCHANGE Peimary /SEconPALY REGISTER CONTENTS STLE ADDITIONAL SITE PARAMETER:
pre, SAND SIZE AT WATER DEPTH oF (.5dy
12 |Excuance Pemarx|secowpary REGISTER CONTENT
CRITICAL. VELOCITY FOR SAND MOTION INITIATION .
_| WaT FOR STATIONARY DISPLAY OF CAINZAL V
CALL SEAwAeD Bounn FuNcTion (S§:).
Guess §,='.2 AnD BEGIN ITERATION,
Wa\T FoR STATIONARY DIS?7LAY oF Ej RooT
Convert § Tod; In METERS AND STOLE,
| Ie pesi@ep , convert dj To Feet
foe NEW CASE, CLEAR Att MEGISTERS
AND RETWEN To Sue ry
Za
STEP KEYENTRY KEY CODE LBL Fa }|32 25 Il 3 03 (6) ae aes Para ae] (ae oro en et = == (a Fs ee RCLIO 34 09 (G7 4+ 07
$T06
h_ RT™ LBL A
EA |
[2] o ~
> vai) ;
Program Listing
COMMENTS STEP KEY ENTRY KEY CODE UNITS CONVERSION ee | CS eee | Oaee| AND STORAGE. Fq@s® & {3i 22 1 as aoe ae eee GTo 0 22 00 a See Ee LS SILA OS EGEEOO | CSTE G@se@) [31 22 24 | Sine | dF uBLo 125 00 eee ae ae (il Rectiomns| M3 4200mma EES ees 1a) (redial ied a Pel aso aaa) 070 ES nieaer A 12 EE ise® 314 eS - $1 (eee ara aa Ty ee) | | STO -O0 ES sieac@ (Ee ea) no se 31 6\ RCL O + 00 Pema IiNWaree esr) Store FUNCTION GNSS] NUMBER, i (ae erage Fimte-piererence | [Ror 6 | 3406 | APPZOMIMATION FOR -—} 7 FIRST DERIWaTive. | | PCL B 34 08 Cees er ae i ee SA Pah eC ares Pe eae) es DEN |RSS 4 eae ae PAS 3) 42 Se aismnOr ass = a) nS [cn | se a (2 ee 32 52 roo] STO 4 | 3304 Sheers RCL4 34 04 (eae ee oe eee) RES ESTOS SSL SS O) SS ieee Se eon GTo 4 22 04 (Sea [a yateicemaiee| [ene Os =] Beebe 2S5s Bac s) x 1 aE ee ee RooT FINDER, iE ADSPa Se aa] REGISTERS 4 *Lolem) [used [2m [Pused [used
COMMENTS
CALCULATE AnD STORE CRITICAL VELOCITY,
LANDWARD BouND FuNCTION.
Sau Hare) [oo md fo cee) |g lemisee} gift /se2)] Bie) | 30.48 Cc D E TPuneti = Si CE) as tele
( (0
STEP KEY ENTRY KEY CODE COMMENTS STEP KEY ENTRY KEY CODE COMMENTS
x So
SEAWARD BOUND PUNCTION,
lw ws)
w N Sa)
ats UP Ir n i
olf —|o N
Baie o|+ WIN
CBS Las
GonNVERT SEAWARD BouND TO METERS.
ay SRY = Bales IN
ra] 33 61 Ob 3406 35 22 32 25 12 | ConvERT LANDWARD BOUND To METERS.
oy ip)
GwvERT Bounds FROM METERS To FEET.
R
Bip o|N @
°o
of 8 o
o} =
Li
< @ a ay
nat ans Pe duel Peedi lm) pueEe nt Ped | __(naction igi Pnctin | vised Fierate [ved
nN Y PL i bs 3
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